3.324 \(\int \sqrt{2+x^2-x^4} \, dx\)

Optimal. Leaf size=44 \[ \frac{1}{3} \sqrt{-x^4+x^2+2} x+F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{1}{3} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]

[Out]

(x*Sqrt[2 + x^2 - x^4])/3 + EllipticE[ArcSin[x/Sqrt[2]], -2]/3 + EllipticF[ArcSi
n[x/Sqrt[2]], -2]

_______________________________________________________________________________________

Rubi [A]  time = 0.130613, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357 \[ \frac{1}{3} \sqrt{-x^4+x^2+2} x+F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{1}{3} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[2 + x^2 - x^4],x]

[Out]

(x*Sqrt[2 + x^2 - x^4])/3 + EllipticE[ArcSin[x/Sqrt[2]], -2]/3 + EllipticF[ArcSi
n[x/Sqrt[2]], -2]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 19.9534, size = 42, normalized size = 0.95 \[ \frac{x \sqrt{- x^{4} + x^{2} + 2}}{3} + \frac{E\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{3} + F\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-x**4+x**2+2)**(1/2),x)

[Out]

x*sqrt(-x**4 + x**2 + 2)/3 + elliptic_e(asin(sqrt(2)*x/2), -2)/3 + elliptic_f(as
in(sqrt(2)*x/2), -2)

_______________________________________________________________________________________

Mathematica [C]  time = 0.0855372, size = 90, normalized size = 2.05 \[ \frac{-x^5+x^3-3 i \sqrt{-2 x^4+2 x^2+4} F\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )+i \sqrt{-2 x^4+2 x^2+4} E\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )+2 x}{3 \sqrt{-x^4+x^2+2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[2 + x^2 - x^4],x]

[Out]

(2*x + x^3 - x^5 + I*Sqrt[4 + 2*x^2 - 2*x^4]*EllipticE[I*ArcSinh[x], -1/2] - (3*
I)*Sqrt[4 + 2*x^2 - 2*x^4]*EllipticF[I*ArcSinh[x], -1/2])/(3*Sqrt[2 + x^2 - x^4]
)

_______________________________________________________________________________________

Maple [B]  time = 0.004, size = 125, normalized size = 2.8 \[{\frac{x}{3}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{2\,\sqrt{2}}{3}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}-{\frac{\sqrt{2}}{6}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1} \left ({\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ) \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^4+x^2+2)^(1/2),x)

[Out]

1/3*x*(-x^4+x^2+2)^(1/2)+2/3*2^(1/2)*(-2*x^2+4)^(1/2)*(x^2+1)^(1/2)/(-x^4+x^2+2)
^(1/2)*EllipticF(1/2*2^(1/2)*x,I*2^(1/2))-1/6*2^(1/2)*(-2*x^2+4)^(1/2)*(x^2+1)^(
1/2)/(-x^4+x^2+2)^(1/2)*(EllipticF(1/2*2^(1/2)*x,I*2^(1/2))-EllipticE(1/2*2^(1/2
)*x,I*2^(1/2)))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{-x^{4} + x^{2} + 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^4 + x^2 + 2),x, algorithm="maxima")

[Out]

integrate(sqrt(-x^4 + x^2 + 2), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{-x^{4} + x^{2} + 2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^4 + x^2 + 2),x, algorithm="fricas")

[Out]

integral(sqrt(-x^4 + x^2 + 2), x)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{- x^{4} + x^{2} + 2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x**4+x**2+2)**(1/2),x)

[Out]

Integral(sqrt(-x**4 + x**2 + 2), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{-x^{4} + x^{2} + 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^4 + x^2 + 2),x, algorithm="giac")

[Out]

integrate(sqrt(-x^4 + x^2 + 2), x)